$\begingroup$Another list of institutes is at fields.utoronto.ca/aboutus/IMSI.html (copied from an answer to a different question on MathOverflow by Brian Borchers); some institutes appear on one list, some on the other, some on both, and some on neither. I expect the list of institutes that I am looking for is quite a bit shorter. $\endgroup$
– Hugh Thomas supports MonicaApr 23 '11 at 9:29

$\begingroup$Oberwolfach recommends applying about six months in advance; a visit can run from two weeks to three months.$\endgroup$
– Hugh Thomas supports MonicaJul 15 '11 at 18:35

$\begingroup$The psu link no longer seems to work. Do you know of an updated link?$\endgroup$
– user62562Oct 13 '17 at 6:48

$\begingroup$According to their website, the "preferred" time to apply (for research in pairs in 2013) is September 30, 2011. However, applications are accepted until four months prior to the desired time. $\endgroup$
– Hugh Thomas supports MonicaJun 17 '11 at 14:35

This is a little bit off the beaten path, and definitely not for all personalities, but the high school summer program Canada/USA Mathcamp has a Research in Pairs program (details here) for which we're happy to receive unsolicited applications. In brief: collaborators visit the program, teach 1 hr/day (on whatever they'd like) to very very bright high school students, and spend the rest of their time on their research program.

$\begingroup$@FanZheng in French "binôme" is used in the mathematical sense, but also generally for designating a group of two. I suppose the double-meaning is what made them chose that word.$\endgroup$
– user9072Oct 7 '15 at 14:25

Collaborate@ICERM offers teams of 3-6 researchers the opportunity to spend five days at the institute during the summer (May-August) or in the month of January. The team research project should have a computational or experimental component.